Describing Data Once we have collected data from surveys or experiments, we need to summarize and present the data in a way that will be meaningful to the reader. We will begin with graphical presentations of data then explore numerical summaries of data. Presenting Categorical Data Graphically Categorical, or qualitative, data are pieces of information that allow us to classify the objects under investigation into various categories. We usually begin working with categorical data by summarizing the data into a frequency table. Frequency Table A frequency table is a table with two columns. One column lists the categories, and another for the frequencies with which the items in the categories occur (how many items fit into each category). Example 1 An insurance company determines vehicle insurance premiums based on known risk factors. If a person is considered a higher risk, their premiums will be higher. One potential factor is the color of your car. The insurance company believes that people with some color cars are more likely to get in accidents. To research this, they examine police reports for recent total-loss collisions. The data is summarized in the frequency table below. Color Blue Green Red White Black Grey Frequency 25 52 41 36 39 23 Sometimes we need an even more intuitive way of displaying data. This is where charts and graphs come in. There are many, many ways of displaying data graphically, but we will concentrate on one very useful type of graph called a bar graph. In this section we will work with bar graphs that display categorical data; the next section will be devoted to bar graphs that display quantitative data. Bar graph A bar graph is a graph that displays a bar for each category with the length of each bar indicating the frequency of that category. To construct a bar graph, we need to draw a vertical axis and a horizontal axis. The vertical direction will have a scale and measure the frequency of each category; the horizontal axis has no scale in this instance. The construction of a bar chart is most easily described by use of an example. Example 2 Using our car data from above, note the highest frequency is 52, so our vertical axis needs to go from 0 to 52, but we might as well use 0 to 55, so that we can put a hash mark every 5 units: 1 Frequency 55 50 45 40 35 30 25 20 15 10 5 0 Blue Green Red White Black Grey Vehicle color involved in total-loss collision Frequency Notice that the height of each bar is determined by the frequency of the corresponding color. The horizontal gridlines are a nice touch, but not necessary. In practice, you will find it useful to draw bar graphs using graph paper, so the gridlines will already be in place, or using technology. Instead of gridlines, we might also list the frequencies at the top of each bar, like this: 55 50 45 40 35 30 25 20 15 10 5 0 52 41 36 39 25 Blue 23 Green Red White Black Grey Vehicle color involved in total-loss collision In this case, our chart might benefit from being reordered from largest to smallest frequency values. This arrangement can make it easier to compare similar values in the chart, even without gridlines. When we arrange the categories in decreasing frequency order like this, it is called a Pareto chart. Pareto chart A Pareto chart is a bar graph ordered from highest to lowest frequency Example 3 Transforming our bar graph from earlier into a Pareto chart, we get: 2 Frequency 52 55 50 45 40 35 30 25 20 15 10 5 0 41 Green Red 39 Black 36 White 25 23 Blue Grey Vehicle color involved in total-loss collision Example 4 In a survey1, adults were asked whether they personally worried about a variety of environmental concerns. The numbers (out of 1012 surveyed) who indicated that they worried \"a great deal\" about some selected concerns are summarized below. Environmental Issue Pollution of drinking water Contamination of soil and water by toxic waste Air pollution Global warming Frequency 597 526 455 354 This data could be shown graphically in a bar graph: 600 Frequency 500 400 300 200 100 0 Water Pollution Toxic Waste Air Pollution Global Warming Environmental Worries To show relative sizes, it is common to use a pie chart. Pie Chart A pie chart is a circle with wedges cut of varying sizes marked out like slices of pie or pizza. The relative sizes of the wedges correspond to the relative frequencies of the categories. Example 5 For our vehicle color data, a pie chart might look like this: 1 Gallup Poll. March 5-8, 2009. http://www.pollingreport.com/enviro.htm 3 Vehicle color involved in total-loss collisions Green Red Black White Blue Grey Pie charts can often benefit from including frequencies or relative frequencies (percents) in the chart next to the pie slices. Often having the category names next to the pie slices also makes the chart clearer. Vehicle color involved in total-loss collisions Grey, 23 Green, 52 Blue, 25 White, 36 Red, 41 Black, 39 Example 6 The pie chart to the right shows the percentage of voters supporting each candidate running for a local senate seat. Voter preferences Ellison 46% Douglas 43% If there are 20,000 voters in the district, the pie chart shows that about 11% of those, about 2,200 voters, support Reeves. Pie charts look nice, but are harder to draw by hand than bar charts since to draw them accurately we would need to Reeves compute the angle each wedge cuts out of the circle, then 11% measure the angle with a protractor. Computers are much better suited to drawing pie charts. Common software programs like Microsoft Word or Excel, OpenOffice.org Write or Calc, or Google Docs are able to create bar graphs, pie charts, and other graph types. There are also numerous online tools that can create graphs 2. Try it Now 1 Create a bar graph and a pie chart to illustrate the grades on a history exam below. A: 12 students, B: 19 students, C: 14 students, D: 4 students, F: 5 students 2 For example: http://nces.ed.gov/nceskids/createAgraph/ or http://docs.google.com 4 Black Grey Car Color White Red Here is another way that fanciness can lead to trouble. Instead of plain bars, it is tempting to substitute meaningful images. This type of graph is called a pictogram. Green 60 50 40 30 20 10 0 Blue Frequency Don't get fancy with graphs! People sometimes add features to graphs that don't help to convey their information. For example, 3-dimensional bar charts like the one shown below are usually not as effective as their two-dimensional counterparts. Pictogram A pictogram is a statistical graphic in which the size of the picture is intended to represent the frequencies or size of the values being represented. Example 7 A labor union might produce the graph to the right to show the difference between the average manager salary and the average worker salary. Looking at the picture, it would be reasonable to guess that the manager salaries is 4 times as large as the worker salaries - the area of the bag looks about 4 times as large. However, the manager salaries are in fact only twice as large as worker salaries, which were reflected in the picture by making the manager bag twice as tall. Manager Worker Another distortion in bar charts results from setting the baseline to a value Salaries Salaries other than zero. The baseline is the bottom of the vertical axis, representing the least number of cases that could have occurred in a category. Normally, this number should be zero. 100 90 80 70 60 50 40 30 20 10 0 60 Frequency (%) Frequency (%) Example 8 Compare the two graphs below showing support for same-sex marriage rights from a poll taken in December 20083. The difference in the vertical scale on the first graph suggests a different story than the true differences in percentages; the second graph makes it look like twice as many people oppose marriage rights as support it. 55 50 45 40 Support Oppose Do you support or oppose same-sex marriage? Support Oppose Do you support or oppose same-sex marriage? 3 CNN/Opinion Research Corporation Poll. Dec 19-21, 2008, from http://www.pollingreport.com/civil.htm 5 Try it Now 2 A poll was taken asking people if they agreed positions of the 4 candidates for a county the pie chart present a good representation of Explain. Nguyen, 42% McKee, 35% with the office. Does this data? Brown, 52% Jones, 64% Presenting Quantitative Data Graphically Quantitative, or numerical, data can also be summarized into frequency tables. Example 9 A teacher records scores on a 20-point quiz for the 30 students in his class. The scores are: 19 20 18 18 17 18 19 17 20 18 20 16 20 15 17 12 18 19 18 19 17 20 18 16 15 18 20 5 0 0 These scores could be summarized into a frequency table by grouping like values: Score 0 5 12 15 16 17 18 19 20 Frequency 2 1 1 2 2 4 8 4 6 Using this table, it would be possible to create a standard bar chart from this summary, like we did for categorical data: 6 Frequency 8 7 6 5 4 3 2 1 0 0 5 12 15 16 17 18 19 20 Score However, since the scores are numerical values, this chart doesn't really make sense; the first and second bars are five values apart, while the later bars are only one value apart. It would be more correct to treat the horizontal axis as a number line. This type of graph is called a histogram. Histogram A histogram is like a bar graph, but where the horizontal axis is a number line Example 10 For the values above, a histogram would look like: 9 8 7 Frequency 6 5 4 3 2 1 0 0 1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 Score Unfortunately, not a lot of software packages can correctly histogram. About the best you Excel or Word is a bar graph with between the bars and spacing simulate a numerical horizontal Frequency Notice that in the histogram, a bar represents values on the horizontal axis from that on the left hand-side of the bar up to, but not including, the value on the right hand side of the bar. Some people choose to have bars start at values to avoid this ambiguity. 8 7 6 5 4 3 2 1 0 common graph a can do in no gap added to axis. 0 2 4 6 8 10 Score 12 14 16 18 20 7 If we have a large number of widely varying data values, creating a frequency table that lists every possible value as a category would lead to an exceptionally long frequency table, and probably would not reveal any patterns. For this reason, it is common with quantitative data to group data into class intervals. Class Intervals Class intervals are groupings of the data. In general, we define class intervals so that: Each interval is equal in size. For example, if the first class contains values from 120-129, the second class should include values from 130-139. We have somewhere between 5 and 20 classes, typically, depending upon the number of data we're working with. Example 11 Suppose that we have collected weights from 100 male subjects as part of a nutrition study. For our weight data, we have values ranging from a low of 121 pounds to a high of 263 pounds, giving a total span of 263121 = 142. We could create 7 intervals with a width of around 20, 14 intervals with a width of around 10, or somewhere in between. Often time we have to experiment with a few possibilities to find something that represents the data well. Let us try using an interval width of 15. We could start at 121, or at 120 since it is a nice round number. Interval 120 - 134 135 - 149 150 - 164 165 - 179 180 - 194 195 - 209 210 - 224 225 - 239 240 - 254 255 - 269 Frequency 4 14 16 28 12 8 7 6 2 3 A histogram of this data would look like: 30 25 Frequency 20 15 10 5 0 120 135 150 165 180 195 210 225 240 255 270 Weights (pounds) 8 In many software packages, you can create a graph similar to a histogram by putting the class intervals as the labels on a bar chart. 30 Frequency 25 20 15 10 5 0 120134 135149 150164 165179 180194 195209 210224 225239 240254 255269 Weights (pounds) Other graph types such as pie charts are possible for quantitative data. The usefulness of different graph types will vary depending upon the number of intervals and the type of data being represented. For example, a pie chart of our weight data is difficult to read because of the quantity of intervals we used. Weights (pounds) 120-134 135-149 150-164 165-179 180-194 195-209 210-224 225-239 240-254 255-269 Try it Now 3 The total cost of textbooks for the term was collected from 36 students. Create a histogram for this data. $140 $285 $330 $160 $160 $285 $290 $340 $345 $165 $300 $350 $180 $300 $355 $220 $305 $360 $235 $310 $360 $240 $310 $380 $250 $315 $395 $260 $315 $420 $280 $320 $460 $285 $320 $460 When collecting data to compare two groups, it is desirable to create a graph that compares quantities. Example 12 The data below came from a task in which the goal is to move a computer mouse to a target on the screen as fast as possible. On 20 of the trials, the target was a small rectangle; on the other 20, the target was a large rectangle. Time to reach the target was recorded on each trial. 9 Interval (milliseconds) 300-399 400-499 500-599 600-699 700-799 800-899 900-999 1000-1099 1100-1199 Frequency small target 0 1 3 6 5 4 0 1 0 Frequency large target 0 5 10 5 0 0 0 0 0 One option to represent this data would be a comparative histogram or bar chart, in which bars for the small target group and large target group are placed next to each other. Frequency 10 8 6 Small Target 4 Large Target 2 0 300399 400499 500599 600699 700799 800899 900- 1000- 1100999 1099 1199 Reaction time (milliseconds) Frequency polygon An alternative representation is a frequency polygon. A frequency polygon starts out like a histogram, but instead of drawing a bar, a point is placed in the midpoint of each interval at height equal to the frequency. Typically the points are connected with straight lines to emphasize the distribution of the data. Example 13 This graph makes it easier to see that reaction times were generally shorter for the larger target, and that the reaction times for the smaller target were more spread out. 10 10 Frequency 8 6 Small Target 4 Large Target 2 0 350 450 550 650 750 850 950 1050 1150 Reaction time (milliseconds) Numerical Summaries of Data It is often desirable to use a few numbers to summarize a distribution. One important aspect of a distribution is where its center is located. Measures of central tendency are discussed first. A second aspect of a distribution is how spread out it is. In other words, how much the data in the distribution vary from one another. The second section describes measures of variability. Measures of Central Tendency Let's begin by trying to find the most "typical" value of a data set. Note that we just used the word "typical" although in many cases you might think of using the word "average." We need to be careful with the word "average" as it means different things to different people in different contexts. One of the most common uses of the word "average" is what mathematicians and statisticians call the arithmetic mean, or just plain old mean for short. "Arithmetic mean" sounds rather fancy, but you have likely calculated a mean many times without realizing it; the mean is what most people think of when they use the word "average". Mean The mean of a set of data is the sum of the data values divided by the number of values. Example 14 Marci's exam scores for her last math class were: 79, 86, 82, 94. The mean of these values would be: 79 86 82 94 85.25 . Typically we round means to one more decimal place than the original data had. 4 In this case, we would round 85.25 to 85.3. Example 15 The number of touchdown (TD) passes thrown by each of the 31 teams in the National Football League in the 2000 season are shown below. 37 33 33 32 29 28 28 23 22 22 22 21 21 21 20 20 19 19 18 18 18 18 16 15 14 14 14 12 12 9 6 Adding these values, we get 634 total TDs. Dividing by 31, the number of data values, we get 634/31 = 20.4516. It would be appropriate to round this to 20.5. 11 It would be most correct for us to report that \"The mean number of touchdown passes thrown in the NFL in the 2000 season was 20.5 passes,\" but it is not uncommon to see the more casual word \"average\" used in place of \"mean.\" Try it Now 4 The price of a jar of peanut butter at 5 stores was: $3.29, $3.59, $3.79, $3.75, and $3.99. Find the mean price. Example 16 The one hundred families in a particular neighborhood are asked their annual household income, to the nearest $5 thousand dollars. The results are summarized in a frequency table below. Income (thousands of dollars) 15 20 25 30 35 40 45 50 Frequency 6 8 11 17 19 20 12 7 Calculating the mean by hand could get tricky if we try to type in all 100 values: 6 terms 15 8 terms 15 20 11 terms 20 25 100 25 We could calculate this more easily by noticing that adding 15 to itself six times is the same as 15 6 = 90. Using this simplification, we get 15 6 20 8 25 11 30 17 35 19 40 20 45 12 50 7 3390 33.9 100 100 The mean household income of our sample is 33.9 thousand dollars ($33,900). Example 17 Extending off the last example, suppose a new family moves into the neighborhood example that has a household income of $5 million ($5000 thousand). Adding this to our sample, our mean is now: 15 6 20 8 25 11 30 17 35 19 40 20 45 12 50 7 5000 1 8390 83.069 101 101 While 83.1 thousand dollars ($83,069) is the correct mean household income, it no longer represents a \"typical\" value. 12 Imagine the data values on a see-saw or balance scale. The mean is the value that keeps the data in balance, like in the picture below. If we graph our household data, the $5 million data value is so far out to the right that the mean has to adjust up to keep things in balance For this reason, when working with data that have outliers - values far outside the primary grouping - it is common to use a different measure of center, the median. Median The median of a set of data is the value in the middle when the data is in order To find the median, begin by listing the data in order from smallest to largest, or largest to smallest. If the number of data values, N, is odd, then the median is the middle data value. This value can be found by rounding N/2 up to the next whole number. If the number of data values is even, there is no one middle value, so we find the mean of the two middle values (values N/2 and N/2 + 1) Example 18 Returning to the football touchdown data, we would start by listing the data in order. Luckily, it was already in decreasing order, so we can work with it without needing to reorder it first. 37 33 33 32 29 28 28 23 22 22 22 21 21 21 20 20 19 19 18 18 18 18 16 15 14 14 14 12 12 9 6 Since there are 31 data values, an odd number, the median will be the middle number, the 16th data value (31/2 = 15.5, round up to 16, leaving 15 values below and 15 above). The 16th data value is 20, so the median number of touchdown passes in the 2000 season was 20 passes. Notice that for this data, the median is fairly close to the mean we calculated earlier, 20.5. Example 19 Find the median of these quiz scores: 5 10 8 6 4 8 2 5 7 7 We start by listing the data in order: 2 4 5 5 6 7 7 8 8 10 Since there are 10 data values, an even number, there is no one middle number. So we find the mean of the two middle numbers, 6 and 7, and get (6+7)/2 = 6.5. The median quiz score was 6.5. 13 Try it Now 5 The price of a jar of peanut butter at 5 stores were: $3.29, $3.59, $3.79, $3.75, and $3.99. Find the median price. Example 20 Let us return now to our original household income data Income (thousands of dollars) 15 20 25 30 35 40 45 50 Frequency 6 8 11 17 19 20 12 7 Here we have 100 data values. If we didn't already know that, we could find it by adding the frequencies. Since 100 is an even number, we need to find the mean of the middle two data values - the 50th and 51st data values. To find these, we start counting up from the bottom: There are 6 data values of $15, so Values 1 to 6 are $15 thousand The next 8 data values are $20, so Values 7 to (6+8)=14 are $20 thousand The next 11 data values are $25, so Values 15 to (14+11)=25 are $25 thousand The next 17 data values are $30, so Values 26 to (25+17)=42 are $30 thousand The next 19 data values are $35, so Values 43 to (42+19)=61 are $35 thousand From this we can tell that values 50 and 51 will be $35 thousand, and the mean of these two values is $35 thousand. The median income in this neighborhood is $35 thousand. Example 21 If we add in the new neighbor with a $5 million household income, then there will be 101 data values, and the 51st value will be the median. As we discovered in the last example, the 51st value is $35 thousand. Notice that the new neighbor did not affect the median in this case. The median is not swayed as much by outliers as the mean is. In addition to the mean and the median, there is one other common measurement of the "typical" value of a data set: the mode. Mode The mode is the element of the data set that occurs most frequently. The mode is fairly useless with data like weights or heights where there are a large number of possible values. The mode is most commonly used for categorical data, for which median and mean cannot be computed. 14 Example 22 In our vehicle color survey, we collected the data Color Blue Green Red White Black Grey Frequency 3 5 4 3 2 3 For this data, Green is the mode, since it is the data value that occurred the most frequently. It is possible for a data set to have more than one mode if several categories have the same frequency, or no modes if each every category occurs only once. Try it Now 6 Reviewers were asked to rate a product on a scale of 1 to 5. Find a. The mean rating b. The median rating c. The mode rating Rating 1 2 3 4 5 Frequency 4 8 7 3 1 Measures of Variation Consider these three sets of quiz scores: Section A: 5 5 5 5 5 5 5 5 5 5 Section B: 0 0 0 0 0 10 10 10 10 10 Section C: 4 4 4 5 5 5 5 6 6 6 All three of these sets of data have a mean of 5 and median of 5, yet the sets of scores are clearly quite different. In section A, everyone had the same score; in section B half the class got no points and the other half got a perfect score, assuming this was a 10-point quiz. Section C was not as consistent as section A, but not as widely varied as section B. In addition to the mean and median, which are measures of the "typical" or "middle" value, we also need a measure of how "spread out" or varied each data set is. 15 There are several ways to measure this "spread" of the data. The first is the simplest and is called the range. Range The range is the difference between the maximum value and the minimum value of the data set. Example 23 Using the quiz scores from above, For section A, the range is 0 since both maximum and minimum are 5 and 5 - 5 = 0 For section B, the range is 10 since 10 - 0 = 10 For section C, the range is 2 since 6 - 4 = 2 In the last example, the range seems to be revealing how spread out the data is. However, suppose we add a fourth section, Section D, with scores 0 5 5 5 5 5 5 5 5 10. This section also has a mean and median of 5. The range is 10, yet this data set is quite different than Section B. To better illuminate the differences, we'll have to turn to more sophisticated measures of variation. Standard deviation The standard deviation is a measure of variation based on measuring how far each data value deviates, or is different, from the mean. A few important characteristics: Standard deviation is always positive. Standard deviation will be zero if all the data values are equal, and will get larger as the data spreads out. Standard deviation has the same units as the original data. Standard deviation, like the mean, can be highly influenced by outliers. Using the data from section D, we could compute for each data value the difference between the data value and the mean: data value 0 5 5 5 5 5 5 5 5 10 deviation: data value - mean 0-5 = -5 5-5 = 0 5-5 = 0 5-5 = 0 5-5 = 0 5-5 = 0 5-5 = 0 5-5 = 0 5-5 = 0 10-5 = 5 We would like to get an idea of the "average" deviation from the mean, but if we find the average of the values in the second column the negative and positive values cancel each other out (this will always happen), so to prevent this we square every value in the second column: data value 0 5 deviation: data value - mean 0-5 = -5 5-5 = 0 deviation squared (-5)2 = 25 02 = 0 16 5 5-5 = 0 02 = 0 5 5-5 = 0 02 = 0 5 5-5 = 0 02 = 0 5 5-5 = 0 02 = 0 5 5-5 = 0 02 = 0 5 5-5 = 0 02 = 0 5 5-5 = 0 02 = 0 10 10-5 = 5 (5)2 = 25 We then add the squared deviations up to get 25 + 0 + 0 + 0 + 0 + 0 + 0 + 0 + 0 + 25 = 50. Ordinarily we would then divide by the number of scores, n, (in this case, 10) to find the mean of the deviations. But we only do this if the data set represents a population; if the data set represents a sample (as it almost always does), we instead divide by n - 1 (in this case, 10 - 1 = 9).4 So in our example, we would have 50/10 = 5 if section D represents a population and 50/9 = about 5.56 if section D represents a sample. These values (5 and 5.56) are called, respectively, the population variance and the sample variance for section D. Variance can be a useful statistical concept, but note that the units of variance in this instance would be points-squared since we squared all of the deviations. What are points-squared? Good question. We would rather deal with the units we started with (points in this case), so to convert back we take the square root and get: 50 population standard deviation 5 2.2 10 or sample standard deviation 50 2.4 9 If we are unsure whether the data set is a sample or a population, we will usually assume it is a sample, and we will round answers to one more decimal place than the original data, as we have done above. To compute standard deviation: 1. Find the deviation of each data from the mean. In other words, subtract the mean from the data value. 2. Square each deviation. 3. Add the squared deviations. 4. Divide by n, the number of data values, if the data represents a whole population; divide by n - 1 if the data is from a sample. 5. Compute the square root of the result. 4 The reason we do this is highly technical, but we can see how it might be useful by considering the case of a small sample from a population that contains an outlier, which would increase the average deviation: the outlier very likely won't be included in the sample, so the mean deviation of the sample would underestimate the mean deviation of the population; thus we divide by a slightly smaller number to get a slightly bigger average deviation. 17 Example 24 Computing the standard deviation for Section B above, we first calculate that the mean is 5. Using a table can help keep track of your computations for the standard deviation: data value 0 0 0 0 0 10 10 10 10 10 deviation: data value - mean 0-5 = -5 0-5 = -5 0-5 = -5 0-5 = -5 0-5 = -5 10-5 = 5 10-5 = 5 10-5 = 5 10-5 = 5 10-5 = 5 deviation squared (-5)2 = 25 (-5)2 = 25 (-5)2 = 25 (-5)2 = 25 (-5)2 = 25 (5)2 = 25 (5)2 = 25 (5)2 = 25 (5)2 = 25 (5)2 = 25 Assuming this data represents a population, we will add the squared deviations, divide by 10, the number of data values, and compute the square root: 25 25 25 25 25 25 25 25 25 25 250 5 10 10 Notice that the standard deviation of this data set is much larger than that of section D since the data in this set is more spread out. For comparison, the standard deviations of all four sections are: Section A: 5 5 5 5 5 5 5 5 5 5 Standard deviation: 0 Section B: 0 0 0 0 0 10 10 10 10 10 Standard deviation: 5 Section C: 4 4 4 5 5 5 5 6 6 6 Standard deviation: 0.8 Section D: 0 5 5 5 5 5 5 5 5 10 Standard deviation: 2.2 Try it Now 7 The price of a jar of peanut butter at 5 stores were: $3.29, $3.59, $3.79, $3.75, and $3.99. Find the standard deviation of the prices. Where standard deviation is a measure of variation based on the mean, quartiles are based on the median. Quartiles Quartiles are values that divide the data in quarters. The first quartile (Q1) is the value so that 25% of the data values are below it; the third quartile (Q3) is the value so that 75% of the data values are below it. You may have guessed that the second quartile is the same as the median, since the median is the value so that 50% of the data values are below it. 18 This divides the data into quarters; 25% of the data is between the minimum and Q1, 25% is between Q1 and the median, 25% is between the median and Q3, and 25% is between Q3 and the maximum value While quartiles are not a 1-number summary of variation like standard deviation, the quartiles are used with the median, minimum, and maximum values to form a 5 number summary of the data. Five number summary The five number summary takes this form: Minimum, Q1, Median, Q3, Maximum To find the first quartile, we need to find the data value so that 25% of the data is below it. If n is the number of data values, we compute a locator by finding 25% of n. If this locator is a decimal value, we round up, and find the data value in that position. If the locator is a whole number, we find the mean of the data value in that position and the next data value. This is identical to the process we used to find the median, except we use 25% of the data values rather than half the data values as the locator. To find the first quartile, Q1 Begin by ordering the data from smallest to largest Compute the locator: L = 0.25n If L is a decimal value: Round up to L+ Use the data value in the L+th position If L is a whole number: Find the mean of the data values in the Lth and L+1th positions. To find the third quartile, Q3 Use the same procedure as for Q1, but with locator: L = 0.75n Examples should help make this clearer. Example 25 Suppose we have measured 9 females and their heights (in inches), sorted from smallest to largest are: 59 60 62 64 66 67 69 70 72 To find the first quartile we first compute the locator: 25% of 9 is L = 0.25(9) = 2.25. Since this value is not a whole number, we round up to 3. The first quartile will be the third data value: 62 inches. To find the third quartile, we again compute the locator: 75% of 9 is 0.75(9) = 6.75. Since this value is not a whole number, we round up to 7. The third quartile will be the seventh data value: 69 inches. Example 26 Suppose we had measured 8 females and their heights (in inches), sorted from smallest to largest are: 59 60 62 64 66 67 69 70 19 To find the first quartile we first compute the locator: 25% of 8 is L = 0.25(8) = 2. Since this value is a whole number, we will find the mean of the 2nd and 3rd data values: (60+62)/2 = 61, so the first quartile is 61 inches. The third quartile is computed similarly, using 75% instead of 25%. L = 0.75(8) = 6. This is a whole number, so we will find the mean of the 6th and 7th data values: (67+69)/2 = 68, so Q3 is 68. Note that the median could be computed the same way, using 50%. The 5-number summary combines the first and third quartile with the minimum, median, and maximum values. Example 27 For the 9 female sample, the median is 66, the minimum is 59, and the maximum is 72. The 5 number summary is: 59, 62, 66, 69, 72. For the 8 female sample, the median is 65, the minimum is 59, and the maximum is 70, so the 5 number summary would be: 59, 61, 65, 68, 70. Example 28 Returning to our quiz score data. In each case, the first quartile locator is 0.25(10) = 2.5, so the first quartile will be the 3rd data value, and the third quartile will be the 8th data value. Creating the five-number summaries: Section and data Section A: 5 5 5 5 5 5 5 5 5 5 Section B: 0 0 0 0 0 10 10 10 10 10 Section C: 4 4 4 5 5 5 5 6 6 6 Section D: 0 5 5 5 5 5 5 5 5 10 5-number summary 5, 5, 5, 5, 5 0, 0, 5, 10, 10 4, 4, 5, 6, 6 0, 5, 5, 5, 10 Of course, with a relatively small data set, finding a five-number summary is a bit silly, since the summary contains almost as many values as the original data. Try it Now 8 The total cost of textbooks for the term was collected from 36 students. Find the 5 number summary of this data. $140 $160 $160 $165 $180 $220 $235 $240 $250 $260 $280 $285 $285 $285 $290 $300 $300 $305 $310 $310 $315 $315 $320 $320 $330 $340 $345 $350 $355 $360 $360 $380 $395 $420 $460 $460 Example 29 Returning to the household income data from earlier, create the five-number summary. 20 Income (thousands of dollars) 15 20 25 30 35 40 45 50 Frequency 6 8 11 17 19 20 12 7 By adding the frequencies, we can see there are 100 data values represented in the table. In Example 20, we found the median was $35 thousand. We can see in the table that the minimum income is $15 thousand, and the maximum is $50 thousand. To find Q1, we calculate the locator: L = 0.25(100) = 25. This is a whole number, so Q1 will be the mean of the 25th and 26th data values. Counting up in the data as we did before, There are 6 data values of $15, so The next 8 data values are $20, so The next 11 data values are $25, so The next 17 data values are $30, so Values 1 to 6 are $15 thousand Values 7 to (6+8)=14 are $20 thousand Values 15 to (14+11)=25 are $25 thousand Values 26 to (25+17)=42 are $30 thousand The 25th data value is $25 thousand, and the 26th data value is $30 thousand, so Q1 will be the mean of these: (25 + 30)/2 = $27.5 thousand. To find Q3, we calculate the locator: L = 0.75(100) = 75. This is a whole number, so Q3 will be the mean of the 75th and 76th data values. Continuing our counting from earlier, The next 19 data values are $35, so Values 43 to (42+19)=61 are $35 thousand The next 20 data values are $40, so Values 61 to (61+20)=81 are $40 thousand Both the 75th and 76th data values lie in this group, so Q3 will be $40 thousand. Putting these values together into a five-number summary, we get: 15, 27.5, 35, 40, 50 Note that the 5 number summary divides the data into four intervals, each of which will contain about 25% of the data. In the previous example, that means about 25% of households have income between $40 thousand and $50 thousand. For visualizing data, there is a graphical representation of a 5-number summary called a box plot, or box and whisker graph. Box plot A box plot is a graphical representation of a five-number summary. To create a box plot, a number line is first drawn. A box is drawn from the first quartile to the third quartile, and a line is drawn through the box at the median. \"Whiskers\" are extended out to the minimum and maximum values. 21 Example 30 The box plot below is based on the 9 female height data with 5 number summary: 59, 62, 66, 69, 72. Example 31 The box plot below is based on the household income data with 5 number summary: 15, 27.5, 35, 40, 50 Try it Now 9 Create a boxplot based on the textbook price data from the last Try it Now. Box plots are particularly useful for comparing data from two populations. Example 32 The box plot of service times for two fast-food restaurants is shown below. 22 While store 2 had a slightly shorter median service time (2.1 minutes vs. 2.3 minutes), store 2 is less consistent, with a wider spread of the data. At store 1, 75% of customers were served within 2.9 minutes, while at store 2, 75% of customers were served within 5.7 minutes. Which store should you go to in a hurry? That depends upon your opinions about luck - 25% of customers at store 2 had to wait between 5.7 and 9.6 minutes. Example 33 The boxplot below is based on the birth weights of infants with severe idiopathic respiratory distress syndrome (SIRDS)5. The boxplot is separated to show the birth weights of infants who survived and those that did not. Comparing the two groups, the boxplot reveals that the birth weights of the infants that died appear to be, overall, smaller than the weights of infants that survived. In fact, we can see that the median birth weight of infants that survived is the same as the third quartile of the infants that died. Similarly, we can see that the first quartile of the survivors is larger than the median weight of those that died, meaning that over 75% of the survivors had a birth weight larger than the median birth weight of those that died. Looking at the maximum value for those that died and the third quartile of the survivors, we can see that over 25% of the survivors had birth weights higher than the heaviest infant that died. The box plot gives us a quick, albeit informal, way to determine that birth weight is quite likely linked to survival of infants with SIRDS. 5 van Vliet, P.K. and Gupta, J.M. (1973) Sodium bicarbonate in idiopathic respiratory distress syndrome. Arch. Disease in Childhood, 48, 249-255. As quoted on http://openlearn.open.ac.uk/mod/oucontent/view.php?id=398296§ion=1.1.3 23 Try it Now Answers 1. D 7% History Exam Grades Frequency 20 F 9% History Exam Grades A 22% 15 10 C 26% 5 0 A B C Grade D F B 36% 2. While the pie chart accurately depicts the relative size of the people agreeing with each candidate, the chart is confusing, since usually percents on a pie chart represent the percentage of the pie the slice represents. 3. Using a class intervals of size 55, we can group our data into six intervals: Cost interval Frequency $140-194 5 $195-249 3 $250-304 9 $305-359 12 $360-414 4 $415-469 3 We can use the frequency distribution to generate the histogram 4. Adding the prices and dividing by 5 we get the mean price: $3.682 5. First we put the data in order: $3.29, $3.59, $3.75, $3.79, $3.99. Since there are an odd number of data, the median will be the middle value, $3.75. 6. There are 23 ratings. 1 4 2 8 3 7 4 3 5 1 a. The mean is 2.5 23 b. There are 23 data values, so the median will be the 12th data value. Ratings of 1 are the first 4 values, while a rating of 2 are the next 8 values, so the 12th value will be a rating of 2. The median is 2. c. The mode is the most frequent rating. The mode rating is 2. 7. Earlier we found the mean of the data was $3.682. 24 data value 3.29 3.59 3.79 3.75 3.99 deviation: data value - mean 3.29 - 3.682 = -0.391 3.59 - 3.682 = -0.092 3.79 - 3.682 = 0.108 3.75 - 3.682 = 0.068 3.99 - 3.682 = 0.308 deviation squared 0.153664 0.008464 0.011664 0.004624 0.094864 This data is from a sample, so we will add the squared deviations, divide by 4, the number of data values minus 1, and compute the square root: 0.153664 0.008464 0.011664 0.004624 0.094864 $0.261 4 8. The data is already in order, so we don't need to sort it first. The minimum value is $140 and the maximum is $460. There are 36 data values so n = 36. n/2 = 18, which is a whole number, so the median is the mean of the 18th and 19th data values, $305 and $310. The median is $307.50. To find the first quartile, we calculate the locator, L = 0.25(36) = 9. Since this is a whole number, we know Q1 is the mean of the 9th and 10th data values, $250 and $260. Q1 = $255. To find the third quartile, we calculate the locator, L = 0.75(36) = 27. Since this is a whole number, we know Q3 is the mean of the 27th and 28th data values, $345 and $350. Q3 = $347.50. The 5 number summary of this data is: $140, $255, $307.50, $347.50, $460 9. Boxplot of textbook costs 25 Exercises Skills 1. The table below shows scores on a Math test. a. Complete the frequency table for the Math test scores b. Construct a histogram of the data c. Construct a pie chart of the data 80 50 50 90 70 70 100 60 70 80 90 100 80 70 30 80 80 70 100 60 70 60 50 50 2. A group of adults where asked how many cars they had in their household a. Complete the frequency table for the car number data b. Construct a histogram of the data c. Construct a pie chart of the data 1 4 2 2 1 2 3 3 1 4 2 2 1 2 1 3 2 2 1 2 1 1 1 2 3. A group of adults were asked how many children they have in their families. The bar graph to the right shows the number of adults who indicated each number of children. a. How many adults where questioned? b. What percentage of the adults questioned had 0 children? 6 Frequency 5 4 3 2 1 0 0 1 2 3 4 5 Number of children Frequency 4. Jasmine was interested in how many days it would take an order from Netflix to arrive at her door. The graph below shows the data she collected. a. How many movies did she order? b. What percentage of the movies arrived in one day? 8 7 6 5 4 3 2 1 0 1 2 3 4 5 Shipping time (days) 5. The bar graph below shows the percentage of students who received each letter grade on their last English paper. The class contains 20 students. What number of students earned an A on their paper? 26 Frequency (%) 40 30 20 10 0 A B C D Number of children 6. Kori categorized her spending for this month into four categories: Rent, Food, Fun, and Other. The percents she spent in each category are pictured here. If she spent a total of $2600 this month, how much did she spend on rent? Fun 16% Other 34% Food 24% Rent 26% 7. A group of diners were asked how much they would pay for a meal. Their responses were: $7.50, $8.25, $9.00, $8.00, $7.25, $7.50, $8.00, $7.00. a. Find the mean b. Find the median c. Write the 5-number summary for this data 8. You recorded the time in seconds it took for 8 participants to solve a puzzle. The times were: 15.2, 18.8, 19.3, 19.7, 20.2, 21.8, 22.1, 29.4. a. Find the mean b. Find the median c. Write the 5-number summary for this data 9. Refer back to the histogram from question #3. a. Compute the mean number of children for the group surveyed b. Compute the median number of children for the group surveyed c. Write the 5-number summary for this data. d. Create box plot. 10. Refer back to the histogram from question #4. a. Compute the mean number of shipping days b. Compute the median number of shipping days c. Write the 5-number summary for this data. d. Create box plot. 27 Concepts 11. The box plot below shows salaries for Actuaries and CPAs. Kendra makes the median salary for an Actuary. Kelsey makes the first quartile salary for a CPA. Who makes more money? How much more? 12. Referring to the boxplot above, what percentage of actuaries makes more than the median salary of a CPA? Exploration 13. Studies are often done by pharmaceutical companies to determine the effectiveness of a treatment program. Suppose that a new AIDS antibody drug is currently under study. It is given to patients once the AIDS symptoms have revealed themselves. Of interest is the average length of time in months patients live once starting the treatment. Two researchers each follow a different set of 40 AIDS patients from the start of treatment until their deaths. The following data (in months) are collected. Researcher 1: 3; 4; 11; 15; 16; 17; 22; 44; 37; 16; 14; 24; 25; 15; 26; 27; 33; 29; 35; 44; 13; 21; 22; 10; 12; 8; 40; 32; 26; 27; 31; 34; 29; 17; 8; 24; 18; 47; 33; 34 Researcher 2: 3; 14; 11; 5; 16; 17; 28; 41; 31; 18; 14; 14; 26; 25; 21; 22; 31; 2; 35; 44; 23; 21; 21; 16; 12; 18; 41; 22; 16; 25; 33; 34; 29; 13; 18; 24; 23; 42; 33; 29 a. Create comparative histograms of the data b. Create comparative boxplots of the data 14. A graph appears below showing the number of adults and children who prefer each type of soda. There were 130 adults and kids surveyed. Discuss some ways in which the graph below could be improved 28 45 40 35 Kids 30 Adults 25 20 Coke Diet Coke Sprite Cherry Coke 15. Make up three data sets with 5 numbers each that have: a. the same mean but different standard deviations. b. the same mean but different medians. c. the same median but different means. 16. A sample of 30 distance scores measured in yards has a mean of 7, a variance of 16, and a standard deviation of 4. You want to convert all your distances from yards to feet, so you multiply each score in the sample by 3. What are the new mean, median, variance, and standard deviation? 17. In your class, design a poll on a topic of interest to you and give it to the class. a. Summarize the data, computing the mean and five-number summary. b. Create a graphical representation of the data. c. Write several sentences about the topic, using your computed statistics as evidence in your writing. 29 Statistics Like most people, you probably feel that it is important to "take control of your life." But what does this mean? Partly it means being able to properly evaluate the data and claims that bombard you every day. If you cannot distinguish good from faulty reasoning, then you are vulnerable to manipulation and to decisions that are not in your best interest. Statistics provides tools that you need in order to react intelligently to information you hear or read. In this sense, Statistics is one of the most important things that you can study. To be more specific, here are some claims that we have heard on several occasions. (We are not saying that any of these claims is true!) o 4 out of 5 dentists recommend Dentyne. o Almost 85% of lung cancers in men and 45% in women are tobacco-related. o Condoms are effective 94% of the time. o Native Americans are significantly more likely to be hit crossing the streets than are people of other ethnicities. o People tend to be more persuasive when they look others directly in the eye and speak loudly and quickly. o Women make 75 cents to every dollar a man makes when they work the same job. o A surprising new study shows that eating egg whites can increase one's life span. o People predict that it is very unlikely there will ever be another baseball player with a batting average over 400. o There is an 80% chance that in a room full of 30 people that at least two people will share the same birthday. o 79.48% of all statistics are made up on the spot. All of these claims are statistical in character. We suspect that some of them sound familiar; if not, we bet that you have heard other claims like them. Notice how diverse the examples are; they come from psychology, health, law, sports, business, etc. Indeed, data and data-interpretation show up in discourse from virtually every facet of contemporary life. Statistics are often presented in an effort to add credibility to an argument or advice. You can see this by paying attention to television advertisements. Many of the numbers thrown about in this way do not represent careful statistical analysis. They can be misleading, and push you into decisions that you might find cause to regret. For these reasons, learning about statistics is a long step towards taking control of your life. (It is not, of course, the only step needed for this purpose.) These chapters will help you learn statistical essentials. It will make you into an intelligent consumer of statistical claims. You can take the first step right away. To be an intelligent consumer of statistics, your first reflex must be to question the statistics that you encounter. The British Prime Minister Benjamin Disraeli famously said, "There are three kinds of lies -- lies, damned lies, and statistics." This quote reminds us why it is so important to understand statistics. So let us invite you to reform your statistical habits from now on. No longer will you blindly accept numbers or findings. Instead, you will begin to think about the numbers, their sources, and most importantly, the procedures used to generate them. We have put the emphasis on defending ourselves against fraudulent claims wrapped up as statistics. Just as important as detecting the deceptive use of statistics is the appreciation of the proper use of statistics. You must also learn to recognize statistical evidence that supports a stated conclusion. When a research team is testing a new treatment for a disease, statistics allows them to conclude based on a relatively small trial that there is good evidence their drug is effective. Statistics allowed prosecutors in the 1950's and 60's to demonstrate racial bias existed in jury panels. Statistics are all around you, sometimes used well, sometimes not. We must learn how to distinguish the two cases. 30 Populations and samples Before we begin gathering and analyzing data we need to characterize the population we are studying. If we want to study the amount of money spent on textbooks by a typical first-year college student, our population might be all first-year students at your college. Or it might be: All first-year community college students in the state of Washington. All first-year students at public colleges and universities in the state of Washington. All first-year students at all colleges and universities in the state of Washington. All first-year students at all colleges and universities in the entire United States. And so on. Population The population of a study is the group the collected data is intended to describe. Sometimes the intended population is called the target population, since if we design our study badly, the collected data might not actually be representative of the intended population. Why is it important to specify the population? We might get different answers to our question as we vary the population we are studying. First-year students at the University of Washington might take slightly more diverse courses than those at your college, and some of these courses may require less popular textbooks that cost more; or, on the other hand, the University Bookstore might have a larger pool of used textbooks, reducing the cost of these books to the students. Whichever the case (and it is likely that some combination of these and other factors are in play), the data we gather from your college will probably not be the same as that from the University of Washington. Particularly when conveying our results to others, we want to be clear about the population we are describing with our data. Example 1 A newspaper website contains a poll asking people their opinion on a recent news article. What is the population? While the target (intended) population may have been all people, the real population of the survey is readers of the website. If we were able to gather data on every member of our population, say the average (we will define "average" more carefully in a subsequent section) amount of money spent on textbooks by each first-year student at your college during the 2009-2010 academic year, the resulting number would be called a parameter. Parameter A parameter is a value (average, percentage, etc.) calculated using all the data from a population We seldom see parameters, however, since surveying an entire population is usually very time-consuming and expensive, unless the population is very small or we already have the data collected. Census A survey of an entire population is called a census. 31 You are probably familiar with two common censuses: the official government Census that attempts to count the population of the U.S. every ten years, and voting, which asks the opinion of all eligible voters in a district. The first of these demonstrates one additional problem with a census: the difficulty in finding and getting participation from everyone in a large population, which can bias, or skew, the results. There are occasionally times when a census is appropriate, usually when the population is fairly small. For example, if the manager of Starbucks wanted to know the average number of hours her employees worked last week, she should be able to pull up payroll records or ask each employee directly. Since surveying an entire population is often impractical, we usually select a sample to study; Sample A sample is a smaller subset of the entire population, ideally one that is fairly representative of the whole population. We will discuss sampling methods in greater detail in a later section. For now, let us assume that samples are chosen in an appropriate manner. If we survey a sample, say 100 first-year students at your college, and find the average amount of money spent by these students on textbooks, the resulting number is called a statistic. Statistic A statistic is a value (average, percentage, etc.) calculated using the data from a sample. Example 2 A researcher wanted to know how citizens of Tacoma felt about a voter initiative. To study this, she goes to the Tacoma Mall and randomly selects 500 shoppers and asks them their opinion. 60% indicate they are supportive of the initiative. What is the sample and population? Is the 60% value a parameter or a statistic? The sample is the 500 shoppers questioned. The population is less clear. While the intended population of this survey was Tacoma citizens, the effective population was mall shoppers. There is no reason to assume that the 500 shoppers questioned would be representative of all Tacoma citizens. The 60% value was based on the sample, so it is a statistic. Try it Now 1 To determine the average length of trout in a lake, researchers catch 20 fish and measure them. What is the sample and population in this study? Try it Now 2 A college reports that the average age of their students is 28 years old. Is this a statistic or a parameter? Categorizing data Once we have gathered data, we might wish to classify it. Roughly speaking, data can be classified as categorical data or quantitative data. 32 Quantitative and categorical data Categorical (qualitative) data are pieces of information that allow us to classify the objects under investigation into various categories. Quantitative data are responses that are numerical in nature and with which we can perform meaningful arithmetic calculations. Example 3 We might conduct a survey to determine the name of the favorite movie that each person in a math class saw in a movie theater. When we conduct such a survey, the responses would look like: Finding Nemo, The Hulk, or Terminator 3: Rise of the Machines. We might count the number of people who give each answer, but the answers themselves do not have any numerical values: we cannot perform computations with an answer like "Finding Nemo." This would be categorical data. Example 4 A survey could ask the number of movies you have seen in a movie theater in the past 12 months (0, 1, 2, 3, 4 ...) This would be quantitative data. Other examples of quantitative data would be the running time of the movie you saw most recently (104 minutes, 137 minutes, 104 minutes, ...) or the amount of money you paid for a movie ticket the last time you went to a movie theater ($5.50, $7.75, $9, ...). Sometimes, determining whether or not data is categorical or quantitative can be a bit trickier. Example 5 Suppose we gather respondents' ZIP codes in a survey to track their geographical location. ZIP codes are numbers, but we can't do any meaningful mathematical calculations with them (it doesn't make sense to say that 98036 is "twice" 49018 that's like saying that Lynnwood, WA is "twice" Battle Creek, MI, which doesn't make sense at all), so ZIP codes are really categorical data. Example 6 A survey about the movie you most recently attended includes the question "How would you rate the movie you just saw?" with these possible answers: 1 - it was awful 2 - it was just OK 3 - I liked it 4 - it was great 5 - best movie ever! Again, there are numbers associated with the responses, but we can't really do any calculations with them: a movie that rates a 4 is not necessarily twice as good as a movie that rates a 2, whatever that means; if two 33 people see the movie and one of them thinks it stinks and the other thinks it's the best ever it doesn't necessarily make sense to say that "on average they liked it." As we study movie-going habits and preferences, we shouldn't forget to specify the population under consideration. If we survey 3-7 year-olds the runaway favorite might be Finding Nemo. 13-17 year-olds might prefer Terminator 3. And 33-37 year-olds might prefer...well, Finding Nemo. Try it Now 3 Classify each measurement as categorical or quantitative a. Eye color of a group of people b. Daily high temperature of a city over several weeks c. Annual income Sampling methods As we mentioned in a previous section, the first thing we should do before conducting a survey is to identify the population that we want to study. Suppose we are hired by a politician to determine the amount of support he has among the electorate should he decide to run for another term. What population should we study? Every person in the district? Not every person is eligible to vote, and regardless of how strongly someone likes or dislikes the candidate, they don't have much to do with him being re-elected if they are not able to vote. What about eligible voters in the district? That might be better, but if someone is eligible to vote but does not register by the deadline, they won't have any say in the election either. What about registered voters? Many people are registered but choose not to vote. What about "likely voters?" This is the criteria used in much political polling, but it is sometimes difficult to define a "likely voter." Is it someone who voted in the last election? In the last general election? In the last presidential election? Should we consider someone who just turned 18 a "likely voter?" They weren't eligible to vote in the past, so how do we judge the likelihood that they will vote in the next election? In November 1998, former professional wrestler Jesse "The Body" Ventura was elected governor of Minnesota. Up until right before the election, most polls showed he had little chance of winning. There were several contributing factors to the polls not reflecting the actual intent of the electorate: Ventura was running on a third-party ticket and most polling methods are better suited to a twocandidate race. Many respondents to polls may have been embarrassed to tell pollsters that they were planning to vote for a professional wrestler. The mere fact that the polls showed Ventura had little chance of winning might have prompted some people to vote for him in protest to send a message to the major-party candidates. But one of the major contributing factors was that Ventura recruited a substantial amount of support from young people, particularly college students, who had never voted before and who registered specifically to vote in the gubernatorial election. The polls did not deem these young people likely voters (since in most cases young people have a lower rate of voter registration and a turnout rate for elections) and so the polling samples were subject to sampling bias: they omitted a portion of the electorate that was weighted in favor of the winning candidate. 34 Sampling bias A sampling method is biased if every member of the population doesn't have equal likelihood of being in the sample. So even identifying the population can be a difficult job, but once we have identified the population, how do we choose an appropriate sample? Remember, although we would prefer to survey all members of the population, this is usually impractical unless the population is very small, so we choose a sample. There are many ways to sample a population, but there is one goal we need to keep in mind: we would like the sample to be representative of the population. Returning to our hypothetical job as a political pollster, we would not anticipate very accurate results if we drew all of our samples from among the customers at a Starbucks, nor would we expect that a sample drawn entirely from the membership list of the local Elks club would provide a useful picture of districtwide support for our candidate. One way to ensure that the sample has a reasonable chance of mirroring the population is to employ randomness. The most basic random method is simple random sampling. Simple random sample A random sample is one in which each member of the population has an equal probability of being chosen. A simple random sample is one in which every member of the population and any group of members has an equal probability of being chosen. Example 7 If we could somehow identify all likely voters in the state, put each of their names on a piece of paper, toss the slips into a (very large) hat and draw 1000 slips out of the hat, we would have a simple random sample. In practice, computers are better suited for this sort of endeavor than millions of slips of paper and extremely large headgear. It is always possible, however, that even a random sample might end up not being totally representative of the population. If we repeatedly take samples of 1000 people from among the population of likely voters in the state of Washington, some of these samples might tend to have a slightly higher percentage of Democrats (or Republicans) than does the general population; some samples might include more older people and some samples might include more younger people; etc. In most cases, this sampling variability is not significant. Sampling variability The natural variation of samples is called sampling variability. This is unavoidable and expected in random sampling, and in most cases is not an issue. To help account for variability, pollsters might instead use a stratified sample. Stratified sampling In stratified sampling, a population is divided into a number of subgroups (or strata). Random samples are then taken from each subgroup with sample sizes proportional to the size of the subgroup in the population. 35 Example 8 Suppose in a particular state that previous data indicated that the electorate was comprised of 39% Democrats, 37% Republicans and 24% independents. In a sample of 1000 people, they would then expect to get about 390 Democrats, 370 Republicans and 240 independents. To accomplish this, they could randomly select 390 people from among those voters known to be Democrats, 370 from those known to be Republicans, and 240 from those with no party affiliation. Stratified sampling can also be used to select a sample with people in desired age groups, a specified mix ratio of males and females, etc. A variation on this technique is called quota sampling. Quota sampling Quota sampling is a variation on stratified sampling, wherein samples are collected in each subgroup until the desired quota is met. Example 9 Suppose the pollsters call people at random, but once they have met their quota of 390 Democrats, they only gather people who do not identify themselves as a Democrat. You may have had the experience of being called by a telephone pollster who started by asking you your age, income, etc. and then thanked you for your time and hung up before asking any "real" questions. Most likely, they already had contacted enough people in your demographic group and were looking for people who were older or younger, richer or poorer, etc. Quota sampling is usually a bit easier than stratified sampling, but also does not ensure the same level of randomness. Another sampling method is cluster sampling, in which the population is divided into groups, and one or more groups are randomly selected to be in the sample. . Cluster sampling In cluster sampling, the population is divided into subgroups (clusters), and a set of subgroups are selected to be in the sample Example 10 If the college wanted to survey students, since students are already divided into classes, they could randomly select 10 classes and give the survey to all the students in those classes. This would be cluster sampling. Other sampling methods include systematic sampling. Systematic sampling In systematic sampling, every nth member of the population is selected to be in the sample. Example 11 To select a sample using systematic sampling, a pollster calls every 100th name in the phone book. 36 Systematic sampling is not as random as a simple random sample (if your name is Albert Aardvark and your sister Alexis Aardvark is right after you in the phone book, there is no way you could both end up in the sample) but it can yield acceptable samples. Perhaps the worst types of sampling methods are convenience samples and voluntary response samples. Convenience sampling and voluntary response sampling Convenience sampling is samples chosen by selecting whoever is convenient. Voluntary response sampling is allowing the sample to volunteer. Example 12 A pollster stands on a street corner and interviews the first 100 people who agree to speak to him. This is a convenience sample. Example 13 A website has a survey asking readers to give their opinion on a tax proposal. This is a self-selected sample, or voluntary response sample, in which respondents volunteer to participate. Usually voluntary response samples are skewed towards people who have a particularly strong opinion about the subject of the survey or who just have way too much time on their hands and enjoy taking surveys. Try it Now 4 In each case, indicate what sampling method was used a. Every 4th person in the class was selected b. A sample was selected to contain 25 men and 35 women c. Viewers of a new show are asked to vote on the show's website d. A website randomly selects 50 of their customers to send a satisfaction survey to e. To survey voters in a town, a polling company randomly selects 10 city blocks, and interviews everyone who lives on those blocks. How to mess things up before you start There are number of ways that a study can be ruined before you even start collecting data. The first we have already explored - sampling or selection bias, which is when the sample is not representative of the population. One example of this is voluntary response bias, which is bias introduced by only collecting data from those who volunteer to participate. This is not the only potential source of bias. Sources of bias Sampling bias - when the sample is not representative of the population Voluntary response bias - the sampling bias that often occurs when the sample is volunteers Self-interest study - bias that can occur when